Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 1, pp 66–76

# Solution of a Multidimensional Tropical Optimization Problem Using Matrix Sparsification

• N. K. Krivulin
• V. N. Sorokin
Mathematics

## Abstract

A complete solution is proposed for the problem of minimizing a function defined on vectors with elements in a tropical (idempotent) semifield. The tropical optimization problem under consideration arises, for example, when we need to find the best (in the sense of the Chebyshev metric) approximate solution to tropical vector equations and occurs in various applications, including scheduling, location, and decision-making problems. To solve the problem, the minimum value of the objective function is determined, the set of solutions is described by a system of inequalities, and one of the solutions is obtained. Thereafter, an extended set of solutions is constructed using the sparsification of the matrix of the problem, and then a complete solution in the form of a family of subsets is derived. Procedures that make it possible to reduce the number of subsets to be examined when constructing the complete solution are described. It is shown how the complete solution can be represented parametrically in a compact vector form. The solution obtained in this study generalizes known results, which are commonly reduced to deriving one solution and do not allow us to find the entire solution set. To illustrate the main results of the work, an example of numerically solving the problem in the set of three-dimensional vectors is given.

## Keywords

idempotent semifield tropical optimization Chebyshev approximation complete solution matrix sparsification

## References

1. 1.
F. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat, Synchronization and Linearity (Wiley, Chichester, 1993), in Ser.: Wiley Series in Probability and Statistics.
2. 2.
V. P. Maslov and V. N. Kolokoltsov, Idempotent Analysis and Its Applications to Optimal Control Theory (Nauka, Moscow, 1994) [in Russian].Google Scholar
3. 3.
R. A. Cuninghame-Green, “Minimax algebra and applications,” in Advances in Imaging and Electron Physics, Ed. by P. W. Hawkes (Academic, San Diego, CA, 1994), Vol. 90, pp. 1–121 https://doi.org/10.1016/S1076-5670(08)70083-1.
4. 4.
J. S. Golan, Semirings and Affine Equations over Them: Theory and Applications (Springer-Verlag, Dordrecht, 2003), in Ser.: Mathematics and Its Applications, Vol. 556. https://doi.org/10.1007/978-94-017-0383-3.Google Scholar
5. 5.
B. Heidergott, G. J. Olsder, and J. van der Woude, Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications (Princeton Univ. Press, Princeton, NJ, 2006), in Ser.: Princeton Series in Applied Mathematics.
6. 6.
M. Gondran and M. Minoux, Graphs, Dioids and Semirings: New Models and Algorithms (Springer-Verlag, New York, 2008), in Ser.: Operations Research / Computer Science Interfaces, Vol. 41. https://doi.org/10.1007/978-0-387-75450-5.Google Scholar
7. 7.
N. K. Krivulin, Methods of Idempotent Algebra for Problems in Modeling and Analysis of Complex Systems (S.-Peterb. Gos. Univ., St. Petersburg, 2009) [in Russian].Google Scholar
8. 8.
K. Glazek, A Guide to the Literature on Semirings and Their Applications in Mathematics and Information Sciences (Springer-Verlag, Dordrecht, 2002). https://doi.org/10.1007/978-94-015-9964-1.
9. 9.
N. N. Vorob’ev, “The extremal algebra of positive matrices,” Elektron. Informationsverarb. Kybernet 3 (1), 39–72 (1967).
10. 10.
R. A. Cuninghame-Green, “Projections in minimax algebra,” Math. Program. 10, 111–123 (1976). https://doi.org/10.1007/BF01580656.
11. 11.
K. Zimmermann, “Some optimization problems with extremal operations,” in Mathematical Programming at Oberwolfach II, Ed. by B. Korte and K. Ritter (Springer-Verlag, Berlin, 1984), in Ser.: Mathematical Programming Studies, Vol. 22, pp. 237–251. https://doi.org/10.1007/BFb0121020.
12. 12.
K. Cechlárová and R. A. Cuninghame-Green, “Soluble approximation of linear systems in max-plus algebra,” Kybernetika 39, 137–141 (2003).
13. 13.
P. Butkovic and K. P. Tam, “On some properties of the image set of a max-linear mapping,” in Tropical and Idempotent Mathematics, Ed. by G. L. Litvinov, S. N. Sergeev (AMS, Providence, RI, 2009), in Ser.: Contemporary Mathematics, Vol. 495, pp. 115–126. https://doi.org/10.1090/conm/495/09694.
14. 14.
N. K. Krivulin, “On solution of linear vector equations in idempotent algebra,” in Mathematical Models. Theory and Applications, Ed. by M. K. Chirkov (VVM, St. Petersburg, 2004), Vol. 5, pp. 105–113 [in Russian].Google Scholar
15. 15.
N. Krivulin, “A new algebraic solution to multidimensional minimax location problems with Chebyshev distance,” WSEAS Trans. Math. 11, 605–614 (2012).Google Scholar
16. 16.
N. Krivulin, “Solution of linear equations and inequalities in idempotent vector spaces,” Int. J. Appl. Math. Inform. 7, 14–23 (2013).Google Scholar
17. 17.
N. Krivulin and K. Zimmermann, “Direct solutions to tropical optimization problems with nonlinear objective functions and boundary constraints,” in Mathematical Methods and Optimization Techniques in Engineering, Ed. by by D. Biolek, H. Walter, I. Utu, and C. von Lucken (WSEAS, 2013), pp. 86–91.Google Scholar
18. 18.
N. Krivulin, “Algebraic solution of tropical optimization problems via matrix sparsification with application to scheduling,” J. Logical Algebraic Methods Program. 89, 150–170 (2017). https://doi.org/10.1016/j.jlamp.2017.03.004.
19. 19.
N. Krivulin, “Extremal properties of tropical eigenvalues and solutions to tropical optimization problems,” Linear Algebra Appl. 468, 211–232 (2015). https://doi.org/10.1016/j.laa.2014.06.044.
20. 20.
N. K. Krivulin and V. N. Sorokin, “Solution of a tropical optimization problem with linear constraints,” Vestn. St. Petersburg Univ. Math. 48, 224–232 (2015). https://doi.org/10.3103/S1063454115040081.